arXiv:2002.04437v1 [math.AG] 11 Feb 2020 · 2020. 2. 12. · galski and J. J. Zhang. This...
Transcript of arXiv:2002.04437v1 [math.AG] 11 Feb 2020 · 2020. 2. 12. · galski and J. J. Zhang. This...
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES, THE MAPS
WHICH HAVE NO INVARIANT PROPER SUBSETS
KEIJI OGUISO AND DE-QI ZHANG
Abstract. Let X be a projective variety and σ a wild automorphism on X , i.e., when-
ever σ(Z) = Z for a Zariski-closed subset Z ofX , we have Z = X . ThenX is conjectured
to be an abelian variety (and proved to be so when dimX ≤ 2) by Z. Reichstein, D. Ro-
galski and J. J. Zhang. This conjecture has been generally open for more than a decade.
In this note, we confirm this conjecture when dimX ≤ 3 and X is not a Calabi-Yau
threefold.
Contents
1. Introduction 1
2. Preliminary results in arbitrary dimensions 3
3. Preliminary results in lower dimensions 9
4. Proof of Theorem 1.2 when q = 2 12
5. Proof of Theorem 1.2 when q = 1 13
6. Proof of Theorem 1.5 17
7. Wild automorphisms of zero entropy 21
References 22
1. Introduction
We work over the complex number field C, unless stated otherwise (cf. Proposition
2.1). Let X be a projective variety. An automorphism σ in Aut(X) is called wild in the
sense of [20] if: whenever σ(Z) = Z for a non-empty Zariski-closed subset Z of X , we
have Z = X . This always holds when dimX = 0.
This note proves the following Theorem 1.2 which supports the following conjecture
([20, Conjecture 0.3]):
2010 Mathematics Subject Classification. 14J50, 32M05, 11G10.
Key words and phrases. Wild automorphism, Abelian variety, Fixed point.
The first named author is partially supported by JSPS and NCTS and the second named author is
partially supported by an ARF of NUS..
1
http://arxiv.org/abs/2002.04437v1
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2 KEIJI OGUISO AND DE-QI ZHANG
Conjecture 1.1. Assume that a projective variety X admits a wild automorphism. Then
X is isomorphic to an abelian variety.
Once X is known to be an abelian variety (as asserted in Conjecture 1.1), the wild
automorphisms on X can been completely classified as in [20, Theorem 0.2]; in particular,
they have zero entropy.
When dimX ≤ 2, Theorem 1.2 is proved in [20, Theorem 6.5] (or Theorem 3.1 for a
slightly simplified proof there). See [10], and also our Section 6 for results when X has
Kodaira dimension zero. Theorem 1.2 follows from Propositions 3.2, 4.1, 5.2 and 6.1.
Theorem 1.2. Let X be a projective variety of dimension ≤ 3 over C. Assume that X
admits a wild automorphism σ. Then either X is an abelian variety, or X is a Calabi
Yau manifold of dimension three and σ has zero entropy.
A projective variety X is a Calabi Yau manifold if its canonical line bundle O(KX) is
torsion and its topological fundamental group π1(Xan) is finite.
Remark 1.3. We believe that a Calabi-Yau threefold X should not have any wild au-
tomorphism σ. This is confirmed if [18, Question 2.6] is affirmative for X and this is
also equivalent to the projective simplicity of the twisted homogeneous coordinate (non-
commutative) ring of any σ-ample line bundle of σ with zero entropy ([20, Proposition
2.2], [9, Theorem 1.2 (2)]); see Section 6 for details.
As is already indicated in Remark 1.3, the motivation of studying wild automorphism
arises from [20, Proposition 0.1] where it is proved that the twisted homogeneous coordi-
nate ring
B(X,L, σ) = ⊕∞n=0H0(X, ⊗n−1i=0 (σ
i)∗L)
associated to any σ-ample line bundle L on X is projective simple if and only if σ is a wild
automorphism on X (cf. [20, Proposition 2.2]); every ample line bundle on an abelian
variety is σ-ample if σ is wild ([20, Corollary 8.6]). More generally, by [9, Theorem 1.2],
X admits a σ-ample line bundle if and only if σ has zero entropy, and in this case, any
ample line bundle is σ-ample. For this reason, the following weaker conjecture would also
be interesting toward Conjecture 1.1 ([20, Conjecture 0.3]).
Conjecture 1.4. Every wild automorphism σ of a projective variety X has zero entropy.
We shall show:
Theorem 1.5. Conjecture 1.4 is true if either one of the following cases occurs.
(1) dimX ≤ 3.
(2) dimX = 4, and the Kodaira dimension κ(X) ≥ 0.
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 3
(3) dimX = 4, κ(X) = −∞, and the irregularity q(X) 6= 1, 2.
Acknowledgement. The authors would like to thank Alfred Jungkai Chen, M. Satriano,
Ch. Schnell and J. Xie for the valuable discussions.
2. Preliminary results in arbitrary dimensions
In this section, we collect some results mostly noticed in [20]. We present them in a
slightly different and expanded way. The main new ingredients are Proposition 2.3 (Fujiki
and Lieberman type result) and Lemma 2.8 on the maximal rationally connected fibration
(a special choice of the birational model: Chow reduction) after [17]. For instance,
Proposition 2.3 can be used to prove Theorem 1.2 for dimX ≤ 2 without any detailed
analysis of the geometry of ruled surfaces. See the proof of Theorem 3.1.
First of all, the following easy remark justifies our assumption on the base field.
Proposition 2.1. Let the base field k be an algebraic closure of a finite field Fq. Then
no automorphism of any projective variety of positive dimension (over k) is wild.
Proof. Suppose the contrary that X is a positive dimensional projective variety over k
and σ is an automorphism of X over k. Since k is an algebraic closure of Fq, both X and
σ are defined over some field k0, which is a finite extension of Fq. In particular, k0 is a
finite field as well. Replacing k0 by a finite extension if necessary, we may assume further
that X(k0) 6= ∅. Here X(k0) is the set of k0-rational points of X . Since k0 is a finite field,
X(k0) is a finite set. Moreover, since σ is defined over k0, we have σ(X(k0)) = X(k0).
Since X(k0) 6= X by dim X > 0, it follows that σ is not wild if dimX > 0. �
Proposition 2.2. (cf. [20, Proposition 3.2, Remark 8.3]) Let X be a projective variety.
Suppose that an algebraic group G ⊆ Aut(X) acts regularly on X such that σ ∈ G is a
wild automorphism. Then X is an abelian variety and σ is a torus translation of infinite
order, with the origin of X chosen in any way.
The result below is proved in [6] and [15] and generalized in [5, Theorem 1.2]. See also
[14, Theorem 1.4] for a purely algebraic proof.
Proposition 2.3. (cf. [6], [15], [5], [14]) Let X be a normal projective variety and we fix
a big Cartier divisor class [ω] ∈ N1(X) of X. Then the group
Aut[ω](X) := {g ∈ Aut(X) | g∗[ω] = [ω]}
is a finite extension of the identity connected component Aut0(X) of Aut(X), that is,
[Aut[ω](X) : Aut0(X)] < ∞.
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4 KEIJI OGUISO AND DE-QI ZHANG
Since Aut0(X) is an algebraic group when X is projective, the above two lemmas with
Lemma 2.5 (2) below imply the following.
Corollary 2.4. Let X be a projective variety with a wild automorphism σ. Suppose a
positive power of σ fixes a big divisor class of X (this is the case when the action of σ on
the Néron Severi group NS(X) is of finite order). Then X is an abelian variety.
We collect some basic properties of wild automorphisms in the two results below.
Lemma 2.5. (cf. [20, Lemma 3.1, Proposition 4.1, Proposition 5.1]) Let X be a projective
variety and let σ be an automorphism on X.
(1) If σ is wild then X is smooth.
(2) σ is wild if and only if so is σm for some m ≥ 1 (and hence for all m ≥ 1).
(3) Suppose that σ is a wild automorphism of X and f : X → Y (resp. g : W →
X with g(Sing (W )) 6= X) is a σ-equivariant surjective morphism of projective
varieties. Then f (resp. g) is a smooth morphism.
(4) Suppose that f : X → Y is a σ-equivariant surjective morphism to a projective
variety Y . If the action of σ on X is wild then so does σ on Y (and hence Y is
smooth).
(5) Suppose that f : X → Y is a σ-equivariant generically finite surjective morphism
of projective varieties. Then the action of σ on X is wild if and only if so does σ
on Y . Further, if this is case, then f : X → Y is a finite étale morphism, and in
particular, it is an isomorphism when f is birational.
(6) If X is an abelian variety and σ is wild then σ has zero entropy.
Proof. (1) Since SingX is stabilized by every automorphism and σ acts as a wild auto-
morphism on X , our X is smooth.
For (2), if σm stabilizes a Zariski-closed subset Z of X , then σ stabilizes the closed
subset ∪m−1i=0 σi(Z) of X . Hence (2) is true.
(3) Let Y1 ⊆ Y be the subset consisting of points y ∈ Y such that the fibre Xy =
X ×k k(y) is not smooth. Let Y2 ⊆ Y be the subset consisting of points y ∈ Y such that
f : X → Y is not flat at y. Since X is smooth, both Yi (resp. their inverses in X) are
Zariski-closed proper subsets of Y (resp. X) and they are σ-stable. Hence Yi = ∅. So f is
smooth. The case of g is similar, by considering the subset of X over which g is non-flat
or singular, each of which is a Zariski-closed subset of W being different from W by the
generic flatness or by our additional assumption that g(Sing (W )) 6= X .
(4) and (5) are similar (and use (3)).
(6) follows from [20, Theorem 0.2]. �
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 5
A smooth projective variety V is a Calabi Yau manifold in the strict sense if V is
simply conencted, KV ∼ 0 and Hj(V,OV ) = 0 (0 < j < dimV ).
Proposition 2.6. (cf. [20, Lemma 3.1, Corollary 4.2, Proposition 4.4, Remark 4.5]) Let
X be a projective variety of dimension ≥ 1, with a wild automorphism σ. Then we have:
(1) Both the Euler Poincaré characteristic and the topological Euler number vanish:
χ(OX) = 0, e(X) = 0. In particular, X is not rationally connected.
(2) Let L ∈ Pic(X) such that σ∗L = L in Pic(X). Then |L| = ∅ or L = OX in
Pic(X). In particular, the Kodaira dimension κ(X) ≤ 0.
(3) Suppose that κ(X) = 0. Then KX ∼Q 0; the Beauville-Bogomolov (minimal split)
finite étale cover X̃ of X is a product of an abelian variety A of dimension ≥ 0
and a few copies of Calabi Yau manifolds Ci of odd dimension ≥ 3 and in the
strict sense; and a positive power of σ lifts to a diagonal action on X̃ = A×∏
i Ci
whose action on each factor is wild.
Proof. (1) See [20, Proposition 4.4, Remark 4.5].
If (2) is false, |L| ≃ PN with N ≥ 1. It follows that the action of σ on |L| has a fixed
point, say D ∈ |L|. Then σ(D) = D, contradicting the fact that σ is wild.
(3) We have KX ∼Q D for some effective Q-divisor (which is unique). Thus D is
σ-stable. Hence D = 0, and KX ∼Q 0. Let X̃ → X be the Beauville-Bogomolov covering
such that X̃ is the product of an abelian variety A, hyperkähler manifolds Hi, and Calabi
Yau manifolds Cj in the strict sense. Replacing the cover X̃ by the minimal splitting
cover in [2, §3], we can lift σ to an action on X̃ so that X̃ → X is σ-equivariant. Hence
the action of σ on X̃ is also wild; further, σ (replaced by a positive power) splits as
diagonal actions on the factors A, Hi and Cj (cf. [2, §3]), which are still wild by Lemma
2.5, and hence these factors have the vanishing Euler Poincare characteristic by (1). Thus
the Hi does not appear, and the Cj are Calabi Yau manifolds of odd dimension (≥ 3) in
the strict sense. This proves the proposition. �
The result below uses the standard results on subvarieties of a complex tori in [23].
Lemma 2.7. (cf. [20, Corollary 4.3, Proposition 5.1]) Let X be a (smooth) projective
variety with a wild automorphism σ, let A be an abelian variety and let f : X → A be a
σ-equivariant morphism. Then the image Y := f(X) is a subtorus of A and f : X → Y is
a smooth surjective morphism. In particular, the albanese map albX : X → A = Alb(X)
is a surjective smooth morphism with connected fibres.
Proof. It is known that every subvariety of an abelian variety, like Y = f(X) of A, has
Kodaira dimension κ(Y ) ≥ 0 and equality holds if and only if Y is a subtorus (after
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6 KEIJI OGUISO AND DE-QI ZHANG
choosing a new origin for A); further, if the inequality is strict then there is a subtorus
B ⊂ A acting on Y such that Y → Y/B is birational to the Iitaka fibration, see [23,
Theorem 10.9, Corollary 10.12]. Now our σ acts on Y and descends to a birational auto-
morphism σ|Y/B on the base of this Iitaka fibration. By the finiteness of the pluricanonical
representation [23, Theorem 14.10], σ|Y/B is of finite order. Hence a sum of finitely many
fibres of X → Y/B is σ-stable. Since σ is wild, this sum equals X itself. Hence Y/B is
a point, so Y equals B, a subtorus of A. The surjective morphism X → Y = f(X) is
smooth by Lemma 2.5.
For the final assertion, note that albX : X → Alb(X) is Aut(X)- and hence σ-
equivariant, and Alb(X) is generated by the image albX(X) (an abelian variety), so
albX(X) = Alb(X). Let g : X → W be the Stein factorization of albX . Since albX is
σ-equivariant, so is g. By Lemma 2.5, the induced morphisms σ|W and σ|Alb(X) are both
wild, W is smooth, and the induced morphism X → W is smooth while W → Alb(X)
is étale. Hence W is also an abelian variety. In fact, the surjective étale morphism
W → Alb(X) is an isomorphism by the universality of the albanese morphism. This
proves the lemma. �
A maximal rationally connected (MRC) fibration on a normal projective variety has
general fibres F rationally connected in the sense of Campana and Kollár-Miyaoka-Mori
(i.e., every two points on F are connected by an irreducible rational curve) and it is
‘maximal’ among such (rational) fibrations. See eg. [11, Definition 5.3, Theorem 5.4].
Lemma 2.8. Let X be a (smooth) projective variety of positive dimension, with a wild
automorphism σ. Suppose that X is uniruled. Then we can choose the maximal rationally
connected (MRC) fibration X → Y to be a well defined σ-equivariant surjective smooth
morphism with 0 < dimY < dimX. Further, the action of σ on Y is also wild.
Proof. By Nakayama [17, Proposition 4.14 or Theorem 4.18], we can choose MRC X 99K
Y to be a (unique) special Chow reduction with the graph Γ = ΓX/Y equi-dimensional
over Y . Especially, σ onX descends to an automorphism σY on Y . The natural birational
map Γ → X is σ-equivariant, and hence it is an isomorphism; see Lemma 2.5. Thus we
may assume that X = Γ → Y is a well-defined surjective morphism. Hence it is smooth
by the same Lemma 2.5. Since X is not rationally connected by Proposition 2.6, Y is not
a point. Since X is uniruled, dimY < dimX . The action of σ on Y is wild by Lemma
2.5. This proves the lemma. �
The following conjecture is known in dimension up to three (cf. [12, §3.13] and the
references therein).
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 7
Conjecture 2.9. (Weak abundance conjecture) A projective variety is uniruled (i.e.,
covered by rational curves) if and only if it has negative Kodaira dimension.
Proposition 2.10. Let X be a (smooth) projective variety of dimension ≥ 1, with a
wild automorphism σ. Assume either dimX ≤ 3 or the weak abundance conjecture in
dimension ≤ n. Then we can choose the maximal rationally connected (MRC) fibration to
be a σ-equivariant surjective smooth morphism f : X → Y (with every fibre a rationally
connected variety of dimension ≥ 0); dimY > 0; the action of σ on Y is wild (and hence
Y is smooth); KY ∼Q 0; and the Beauville-Bogomolov finite étale cover of Y is a product
of an abelian variety of dimension ≥ 0 and a few copies of Calabi-Yau manifolds of odd
dimension ≥ 3 in the strict sense.
Proof. By Proposition 2.6, we have the Kodaira dimension κ(X) ≤ 0. If κ(X) = 0, the
proposition follows from Proposition 2.6.
Thus we may assume that κ(X) < 0. Then X is uniruled, by our assumption. Using
Lemma 2.8, we can choose the maximal rationally connected (MRC) fibration X → Y to
be a well defined σ-equivariant surjective smooth (non-trivial) morphism. By Lemma 2.5
the induced action σ|Y on Y of σ is also wild. Thus every fibre of X → Y is rationally
connected, since σ|Y fixes the locus on Y (a non-dense proper subset of Y ) over which
the fibres are not rationally connected. Y is not uniruled by [7, Corollary 1.4]. Hence
the Kodaira dimension κ(Y ) ≥ 0 by the assumed weak abundance conjecture. Now the
proposition follows from Proposition 2.6. �
Lemma 2.11. Let X be a (smooth) projective variety with a wild automorphism σ, let
A be an abelian variety and let f : X → A be a σ-equivariant surjective morphism with
connected fibres F of positive dimension. Suppose that σ fixes the class of a big divisor
D on A in NSR(A) = NS(A)⊗Z R (this holds if dimA = 1 or a positive power of σ is a
translation on A). Then −KF is not a big divisor.
Proof. By Lemma 2.5, f is a smooth morphism. Suppose the contrary that −KF is a big
divisor. Then −KX is relatively big over A. Thus E := −KX + mf∗D is a big divisor
for some m >> 1, see (1) in the proof of [3, Lemma 2.5]. Now σ fixes the big class
[E]. Thus, by Corollary 2.4, X is an abelian variety and hence KF = (KX)|F = 0, a
contradiction. This proves the lemma. Indeed, if dimA = 1, then σ is a translation on
A by [20, Theorem 0.2]. If σ1 = σt is a translation on A then for any ample divisor H
on A we have the class of H fixed by σ1, hence σ fixes the class of the ample divisor
D :=∑t−1
i=0(σi)∗H . �
The lemma below was proved in [10].
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8 KEIJI OGUISO AND DE-QI ZHANG
Lemma 2.12. (cf. [10, Lemma 2.8]) Let σ be a wild automorphism of an abelian variety
A. Let H ≤ Autvariety(A) be a finite subgroup centralized by σ: σh = hσ for all h ∈ H.
Then H consists of translations of A.
A projective variety X is called a Q-torus if it has an abelian variety A1 as an étale
finite cover (or equivalently it is the quotient of an abelian variety A2 by a finite group
acting freely on A).
Proposition 2.13. Let X be a projective Q-torus with a wild automorphism σ. Then X
is an abelian variety.
Proof. Let A → X be the minimal splitting cover of X in [2, §3]. Then σ lifts to an
automorphism on A, also denoted as σ. Note that the σ on A normalizes H := Gal(A/X).
Hence σr! centralizes every element of H , where r := |H|. Since σr! is still wild by Lemma
2.5, H consists of translations by Lemma 2.12. Hence H = {idA} by the minimality of
A → X . Therefore X = A and X is an abelian variety. �
Lemma 2.14. Let X be a smooth projective variety, C a smooth projective curve, and
f : X → C a smooth morphism. Let η ∈ C be the generic point in the sense of scheme
and Xη = f−1(η) the generic fibre. Suppose Pic(Xη) ≃ NS(Xη) (this is the case when
Pic0(Xη) = 0 , eg. when the geometric generic fibre Xη is rationally connected). Then
NS(X)|Xη = NS(Xη) and the kernel of the surjection
NSQ(X) → NSQ(X)|Xη = NSQ(Xη)
is spanned by Q[F ], where [F ] is the class of closed fibres of f .
Proof. Note that algebraic equivalence is preserved under morphisms (hence under re-
strictions). Thus NS(X)|Xη ⊆ NS(Xη). Let D be a prime divisor on Xη and let D be the
Zariski closure of D in X . Then D is a divisor on X and D = D|Xη as divisors. This
implies NS(X)|Xη = NS(Xη).
Let L,M ∈ Pic(X) such that L|Xη∼= M|Xη in Pic(Xη). It suffices to show that
L = M ⊗ f ∗N for some N ∈ Pic(C).
By L|Xη∼= M|Xη , it follows that
h0(Xη, (L⊗M−1)|Xη) = 1 , h
0(Xη, (L⊗M−1)−1|Xη) = 1 .
Then by the upper-semicontinuity theorem,
h0(Xy, (L⊗M−1)|Xy) ≥ 1 , h
0(Xy, (L⊗M−1)−1|Xy) ≥ 1
for all closed fibres Xy. Since Xy is smooth projective, it follows that (L×M−1)|Xy ≃ OXy
and therefore h0(Xy, (L×M−1)|Xy) = 1 for all closed fibres Xy. Then, since the morphism
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 9
f is smooth, hence flat, the sheaf N := f∗(L ⊗ M−1) is an invertible sheaf with base
change property f∗(L ⊗ M−1) ⊗ κ(y) ≃ H0(Xy, (L × M
−1)|Xy) by Grauert’s theorem.
Since (L ⊗ M−1)|Xy ≃ OXy is globally generated, by Nakayama’s lemma, the natural
morphism
f ∗N := f ∗f∗(L⊗M−1) → L⊗M−1
is then surjective. Since the both sides are invertible sheaves, it follows that f ∗N ≃
L⊗M−1, i.e., L = M ⊗ f ∗N as desired. �
3. Preliminary results in lower dimensions
Theorem 3.1 below was proved in [20, §6]. Our Lemma 2.11 (and Proposition 2.3
implicitly) helps skipping the detailed analysis of ruled surfaces; compare with the proof
of [20, Lemma 6.2].
Theorem 3.1. (cf. [20, §6]) Let X be a projective variety with a wild automorphism σ.
Suppose that dimX ≤ 2. Then X is an abelian variety.
Proof. Assume to the contrary that X is not an abelian variety. Then by Proposition
2.10 and Proposition 2.13, it follows that dimX = 2 and X admits a smooth fibration
f : X → Y with fibres F smooth rational curve and Y an elliptic curve. But then F ∼= P1
has ample −KF , contradicting Lemma 2.11. �
Theorem 1.2 for threefolds will follow from Proposition 3.2 below and Propositions 4.1,
5.2 and 6.1 in subsequence sections.
Proposition 3.2. Let X be a projective variety of dimension three with a wild automor-
phism σ. Suppose that X is neither an abelian variety, nor a Calabi-Yau manifold. Then
X is uniruled and the maximal rationally connected (MRC) fibration can be chosen to be
a σ-equivariant surjective smooth morphism f : X → Y and coincides with the albanese
map albX : X → Alb(X) = Alb(Y ), such that σ acts on Y as a wild automorphism and
one of the following cases occurs.
(1) The irregularity q(X) = 1. In this case, Y is an elliptic curve, σ acts on Y as a
translation of infinite order and every fibre Xy over y ∈ Y is a smooth rational
surface.
(2) q(X) = 2. In this case, Y is an abelian surface, σ acts on Y as a wild automor-
phism, and every fibre Xy over y ∈ Y is a smooth rational curve: Xy ∼= P1.
Proof. Since X is neither an abelian threefold, nor a Calabi-Yau threefold, by virtue of
Proposition 2.10 and Proposition 2.13, X is uniruled and the MRC f : X → Y is a
σ-equivariant surjective smooth morphism with Y an abelian variety of dimension 1 or
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10 KEIJI OGUISO AND DE-QI ZHANG
2 and every fibre F a smooth rational variety of dimension 2 or 1, respectively. Here we
use the fact that a rationally connected variety of dimension at most two is a rational
variety.
Since Y = Alb(Y ), there is a morphism g : Alb(X) → Y such that g ◦ albX = f by
the universal property of the albanese map albX . Since f is surjective with connected
fibres, so is g. Let Xy (y ∈ Y ) be any closed fibre of f . Then albX(Xy) is a point for
each Xy, because Xy is rationally connected and any abelian variety contains no rational
curves. Thus g is a finite morphism (with connected fibres), so it is an isomorphism by the
normality (and indeed smoothness) of Alb(X) and Y . This proves the proposition. �
The next result is certainly known at least over C (cf. [21, Table 3.1]).
Proposition 3.3. Let π : S → E be a relatively minimal elliptic ruled surface over an
algebraically closed field of characteristic 6= 2. Then κ(S,−KS) ≥ 0.
Proof. We write S = P(E) = Proj(⊕n≥0Symn(E)) where E is a normalized rank 2 locally
free sheaf on the elliptic curve E. Then we have an exact sequence
0 → OE → E → L → 0
for some invertible sheaf L on E. Let C0 be the section corresponding to this surjection.
Then, with respect to OS(1) defined by E , one has OS(1) = OS(C0) and
−KS = OS(2)− π∗det E = O(2C0)− π
∗det E = O(2C0)− π∗L
in Pic(S) by the canonical bundle formula.
If E is decomposable, then degL ≤ 0 and therefore E = OE ⊕ L. Corresponding to
the projection E → OE , we have a section C1 such that OS(C1) = OP(E)(1)− π∗det E in
Pic(S). Hence −KS is linearly equivalent to an effective divisor C0 + C1.
If E is indecomposable of degree 0, then E is the non-trivial extension of the form
0 → OE → E → OE → 0
and therefore −KS ∼= OS(2). Hence −KS is linearly equivalent to the effective divisor
2C0, where C0 is the section given by the exact sequence above.
It remains to consider the case where E is indecomposable of degree 1. Our proof below
is inspired by [22].
Consider the product surface P1 × E and its automorphism subgroup G generated by
τ1 and τ2 defined by
τ1 : (t, s) 7→ (−t, s + a), τ2 : (t, s) 7→ (t−1, s+ b)
where 〈a, b〉 is the group of two torsion points of E. Then G acts freely on P1 × E and
we have a smooth projective surface S = (P1 × E)/G. The two projections on P1 × E
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 11
induce on S a relatively minimal elliptic fibre space structure p1 : S → P1 = P1/G and a
relatively minimal elliptic ruled surface structure p2 : S → E ≃ E/G.
We are going to show that κ(S,−KS) ≥ 0 and p2 : S → E ≃ E/G is the unique
minimal elliptic ruled surface over E corresponding to an indecomposable locally free
sheaf of rank 2 and degree 1.
Observe that p1 has exactly three multiple fibres of type 2I0. Hence, by the canonical
bundle formula for an elliptic surface applied for p1, it follows that −2KS is linearly
equivalent to F , any smooth fibre of p1. In particular, κ(S,−KS) ≥ 0.
Now it suffices to show that p2 : S → E is a ruled surface corresponding to an indecom-
posable locally free shaef E of rank 2 and degree 1. Recall that such a minimal elliptic
ruled surface is unique from E and does not depend on the choice of such sheaf E , and
this is also the unique case so that (C20) > 0, where C0 is the section of p2 as defined at
the beginning of this proof.
Observe that deg p2|Ft = 2 or 4 for any fibre Ft of p1 with reduced structure. Then the
section C0 of p2 is an elliptic curve, but not in the fibre of p1 (whose multiplicity is at
most 2). Thus (F.C0) > 0 and hence (C20) > 0 by
0 = degKC0 = ((KS + C0).C0) = −(F.C0)/2 + (C20) .
This proves the result. �
Lemma 3.4. Let π : S → E be a relatively minimal ruled surface over an elliptic curve
E. Then κ(S,−KS) ≥ 0 and one of the following cases occurs.
(1) κ(S,−KS) = 2. In this case for all t >> 1 and sufficiently divisible, the base locus
|−tKS| has 1-dimensional part.
(2) κ(S,−KS) = 1. In this case, |−tKS| is base point free for some t ≥ 1 and it
defines a relatively minimal elliptic fibration S → P1.
(3) κ(S,−KS) = 0. In this case, there is m > 0 such that |−mKS| = {D} for a
non-zero effective divisor D on S.
Proof. Note that κ(S,−KS) ≥ 0 by Proposition 3.3 or [21, Table 3.1].
Let −KS = P+N be the Zariski-decomposition, where P is nef, P.N = 0, N =∑
aiNi
is effective and has negative intersection matrix (Ni.Nj). Note that κ(S,−KS) = κ(S, P ),
and the integral part of tN is contained in the fixed part of |−tKS| for all t ≥ 1.
If κ(S,−KS) = 2, then −KS and hence P are big. So P is nef and big, and P2 > 0.
Thus 0 = (−KS)2 = P 2 +N2 > N2 implies that N 6= 0. This is Case(1).
Now we may assume that κ(S,−KS) = 0 or 1. Hence P is not big, so P2 = 0. Thus
0 = (−KS)2 = P 2 +N2 = N2 implies N = 0 (the zero divisor). Namely, −KS = P is nef
(but not big).
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12 KEIJI OGUISO AND DE-QI ZHANG
Suppose that κ(S,−KS) = 0. Then the statement in Case (3) is clear, because −KS ,
hence KS, is not numerically trivial.
Suppose that κ(S,−KS) = 1. Since −KS is nef and (−KS)2 = 0, it follows that |−tKS|
is base point free for t >> 1 and sufficiently divisible. The adjunction formula for fibre,
implies then that |−tKS| defines an elliptic fibration f : S → P1 (with fibres of the ruling
S → E all horizontal to f). This proves the lemma. �
4. Proof of Theorem 1.2 when q = 2
Proposition 4.1. The case q(X) = 2 in Proposition 3.2 does not occur.
Proof. Suppose the contrary that the case q(X) = 2 in Proposition 3.2 occurs and we
shall derive a contradiction.
For the σ-equivariant fibration f : X → Y there, let Xy be a fibre over y ∈ Y . Then
Xy ∼= P1. As in [20, Theorem 7.2], write the wild automorphism σ = Tb ◦ α on Y where
Tb is a translation on the abelian surface Y and α : Y → Y is a group automorphism
such that the endomorphism β = α− idY is nilpotent.
If β = 0 then σ = Tb and it is a translation; by Lemma 2.11, the anti-canonical divisor
−KXy is not big, which contradicts that Xy ∼= P1 has ample anti-canonical divisor.
Thus we may assume that β 6= 0. Let B be a connected component of Kerβ. Then
B is a nontrivial proper subtorus of the 2-torus Y , and hence an elliptic curve. Thus σ
permutes the coset of E := Y/B, an elliptic curve. Hence the quotient map Y → E is
σ-equivariant. Since the action of σ on Y is wild, so does σ on E, see Lemma 2.5. Hence
σ is a translation of infinite order, see [20, Remark 8.3].
Consider the σ-equivariant fibration g : X → E, which is the composition X → Y →
E = Y/B. It is a smooth fibration (cf. Lemma 2.5) with each fibre Xe over e ∈ E
a relatively minimal ruled surface over the elliptic curve Ye (the fibre of Y → E over
e ∈ E), noting that every fibre of X → Y is isomorphic to P1.
For −KXe = (−KX)|Xe, note that h0(Xe,O(−mKX)|Xe) and hence κ(Xe,−KXe) are
upper semi-continuous as functions in e ∈ E. The subsets of points of E over which
these functions obtain larger values, are Zariski closed subsets and σ-stable. Since σ is
wild, we may assume that these functions are all constants as functions in e ∈ E. Since
g : X → E (a smooth curve) is flat and by Grauert’s theorem, g∗O(−mKX) is locally
free and we have the natural isomorphism for all e ∈ E and m ≥ 0
(∗) g∗O(−mKX)⊗ k(e) ∼= H0(Xe,O(−mKX)|Xe).
For m >> 1 and sufficiently divisible, consider the rational map
h : X 99K W = P(g∗O(−mKX))
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 13
associated to the sheaf homomorphism
g∗g∗O(−mKX) → O(−mKX) .
Since σ∗(−KX) = −KX , there is a natural action of σ on W compatible with that on X .
If S = Xe has κ(S,−KS) = 2 or 0, then Lemma 3.4 and the isomorphism (∗) above
imply that Bs |−mKS| is 1-dimensional for some m >> 1 and divisible, whose union
∪e∈E Bs |−mKXe | = SuppCoker (g∗g∗O(−mKX) → O(−mKX))
is a σ-stable Zariski-closed proper subset of X , a contradiction.
If S = Xe has κ(S,−KS) = 1, then Lemma 3.4 and the isomorphism (*) above imply
that |−mKS| defines an elliptic fibration S → P1 for some m >> 1 and divisible, and
h : X → W is a well-defined morphism. Thus W is a ruled surface, covered by rational
curves h(Xe), and we have a σ-equivariant fibration h : X → W . But the action of σ on
W is also wild by Lemma 2.5, and hence W must be an abelian surface by [20, §6] or
Theorem 3.1. This is a contradiction. This proves the proposition. �
5. Proof of Theorem 1.2 when q = 1
We begin with:
Lemma 5.1. Let S be a smooth projective rational surface over a field k (not necessarily
algebraically closed) and Λ := NS(S/k), the Néron-Severi group of S over the field k. Let
g ∈ Aut(X/k) be an automorphism of S over k. Assume that g is of zero entropy in the
sense that the spectral radius of g∗|Λ is one, and g∗|Λ is of infinite order. Then we have:
(1) There is a nonzero pseudo-effective Q-Cartier divisor v ∈ ΛQ := Λ⊗ZQ such that
g(v) ∼Q v. Moreover, the ray Q>0v of such v is unique.
(2) v is nef and v2 = 0.
(3) Replacing v by a multiple, we have
(∗) KS + v ∼Q N(S) :=∑
i
aiNi.
Here if N(S) 6= 0, then N(S) is an effective divisor with Ni integral curves, the
intersection matrix (Ni.Nj) is negative definite and g(N(S)) ∼Q N(S). The v in
(1) with the extra condition (*) above is unique up to Q-linear equivalence.
Proof. Since Λ is hyperbolic, (1) and (2) are proved in the exactly same way as in [19,
Theorem 2.1], noting that q(S) = 0 and hence numerical equivalence coincides with
Q-linear equivalence on NS(S/k) = Pic(S/k).
For (3), by replacing v by a positive multiple, we may assume that v ∈ Λ = Pic(S) is
an integral Cartier divisor.
-
14 KEIJI OGUISO AND DE-QI ZHANG
First note that KS.v = 0. Otherwise, (KS + tv)2 > 0 for some integer t and g∗(KS +
tv) = KS + tv. Since (KS + tv)⊥ (⊂ Λ) is of negative definite, it follows that g∗|Λ is of
finite order, a contradiction.
By the Serre duality, we also have
h2(KS + v) = h0(−v) = 0
as v is nef ad v 6= 0. Hence, by the Riemann-Roch formula, we obtain
h0(KS + v) ≥(KS + v).v
2+ χ(OS) = 1.
If KS + v = 0 in Pic(S), then (3) clearly holds. If KS + v 6= 0 in Pic(S), then one can
take the Zariski decomposition
KS + v = P +N
in ΛQ (see eg. [1, Theorem] whose proof is valid over any field k). Now the same
proof of [25, Lemma 4.3, Page 175] implies the result. Indeed, g∗(P ) = P in ΛQ by
g∗(KS + v) = KS + v and by the uniqueness of the Zariski decomposition. Then, by (1),
P = av in ΛQ for some rational number a. The rational number 1 − a ∈ Q is positive,
since a smooth rational surface S has KS non-pseudo effective. Replacing v by (1− a)v
and putting N(S) := N , we obtain (3). Here the equality g∗N = N as Q-divisors follows
from g∗N = N in ΛQ, as the matrix (Ni.Nj) is negative definite (by the definition of the
Zariski decomposition) . �
Proposition 5.2. The case q(X) = 1 in Proposition 3.2 does not occur.
Proof. Suppose the contrary that the case q(X) = 1 in Proposition 3.2 occurs, we shall
derive a contradiction.
For the σ-equivariant fibration f : X → Y there, let Xy be a fibre over y ∈ Y . Then
F = Xy is a smooth rational surface and Y is an elliptic curve. For the wild automorphism
action of σ on the elliptic curve Y , we have that σ is a translation of infinite order.
Let η ∈ Y be the scheme generic point of Y and Xη = f−1(η) the scheme generic fibre
of f . Recalling that NS(X)|Xη = NS(Xη), we consider the natural action σ∗ on
Λ := NS(X)|Xη = NS(Xη)
defined as:
σ∗(M|Xη) := (σ∗M)|Xη .
We can also define a pairing on Λ by
(M1|Xη .M2|Xη)Xη .
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 15
This paring is compatible with that on NS(Xη). It is non-degenerate and has signature
(1, r− 1) with r = rankΛ ≤ rankNS(Xη), since H|Xη 6= 0 in Λ for an ample divisor H on
X . Namely, Λ is a hyperbolic lattice.
By the flatness of f , we have
(M1|Xη .M2|Xη)Xη = (M1|F .M2F )F = (M1.M2.F )X .
Here F is any closed fibre of f .
Since σ∗F ≡ F (numerical equivalence), we have
(σ∗(M1|Xη).σ∗(M2|Xη)) = (σ
∗M1.σ∗M2.F )X = (M1.M2.F )X = (M1|Xη .M2|Xη).
Hence the action of σ on Λ is an isometry with respect to its pairing and the spectral radius
ρ(σ∗ |Λ) is just the first relative dynamical degree d1(σ|f ) defined in [4]. In particular, we
have
d1(σ) = ρ(σ∗|Λ)
by the product formula, because Y is a curve with d1(σ|Y ) = 1.
Run the relative minimal model program (MMP) over the elliptic curve Y . LetX → X1
be the contraction of a KX -negative extremal ray over Y . Then ρ(X) = ρ(X1) + 1.
Suppose that the contraction X → X1 over Y is a Fano contraction, i.e., a Mori fibre
space. Note that every variety has only finitely many Fano contractions of extremal rays
(cf. [24, Theorem 2.2]). Replacing σ by a positive power, we may assume that σ stabilizes
the extremal ray of the contraction X → X1 and hence the action of σ descends to that
on X1 (cf. e.g. [26, Lemma 2.12]) which is again wild, see Lemma 2.5.
If dimX1 ≤ 1, then X1 = Y , the fibration X → Y is a Fano contraction and ρ(X) = 2.
Now NSQ(X) is spanned by −KX and a fibre Xy over y ∈ Y both classes of which are
σ-stable. So σ∗ |NS(X) is of finite order. Thus X is an abelian variety by Corollary 2.4,
contradicting the existence of a Fano contraction X → Y .
If dimX1 = 2, then X1 is an abelian surface by Corollary 3.1. However, then 1 =
q(X) ≥ q(X1) = 2, a contradiction.
Thus we may assume that X → X1 is birational. Since a smooth threefold has no
flip, X → X1 is a divisorial contraction with the exceptional locus D0, a prime divisor,
see [16]. Since we are running the relative MMP of X over the elliptic curve Y , the
intersection of D0 with each fibre of X → Y is a union of several (rational) curves to be
contracted by the map X → X1.
If the exceptional divisor D0 is σ-periodic, the union of σi(D0) (i ≥ 0) is a σ-stable
Zariski-closed proper subset of X , a contradiction. Therefore, D0 is not σ-periodic.
Now the proposition follows from the three claims below.
Claim 5.3. d1(σ|X) = 1.
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16 KEIJI OGUISO AND DE-QI ZHANG
Proof. Suppose the contrary that d1(σ|X) > 1. Then, since the locus D0 of the divisorial
extremal contraction X → X1 is not σ-periodic, [13, Lemma 6.3, Theorem 1.7] proves the
lifting of σ to some X ′ with a σ-equivariant birational morphism ρξ : X′ → X , and the
descending of σ to some surface S with a σ-equivariant surjective morphism τ : X ′ → S.
Both the action of σ on X ′ and S are wild; see Lemma 2.5. Now S is an abelian surface
by Corollary 3.1. However, then 1 = q(X) = q(X ′) ≥ q(S) = 2, a contradiction. Thus
the claim is true. �
From now on, we may assume that ρ(σ∗|Λ) = d1(σ|X) = 1.
Claim 5.4. The action of σ on the hyperbolic lattice Λ is of infinite order.
Proof. Suppose the contrary that that σ∗|Λ is of finite order. Replacing σ by a positive
power, we may assume that σ∗ acts on Λ as the identity. Note that σ−1D0 is the excep-
tional locus of another divisorial contraction of extremal ray on X , see e.g. [26, Lemma
2.11]. Let
Xη = Xη ×Spec κ(η) Specκ(η)
be the geometric generic fibre of f . Since both D0|Xη and (σ−1D0)|Xη are defined over
Xη, it follows from σ∗ = id on Λ (⊂ NS(Xη)) that
(D0)|Xη = (σ∗D0)|Xη ∈ NS(Xη)
On the other hand, both (D0)|Xη and (σ∗D0)|Xη are unions of a few contractible curves
on the smooth rational projective surface Xη. The negativity of D0|Xη implies that
D0|Xη = (σ−1(D0))|Xη as sets. Then D0|Xη = (σ
−1(D0))|Xη as divisors on Xη. By taking
the Zariski closure, we obtain D0 = σ−1(D0), as both D0 and σ
−1D0 are irreducible
divisors on X . However, this contradicts the fact that D0 is not σ-periodic (σ being
wild). This proves the claim. �
To finish the proof of Proposition 5.2, we still have to prove the following.
Claim 5.5. It is impossible that the action of σ on the hyperbolic lattice Λ is parabolic,
i.e., it is of infinite order and ρ(σ∗|Λ) = d1(σ|X) = 1.
Proof. Suppose the contrary that the action of σ on Λ is parabolic. We apply Lemma
5.1 to Case q(X) = 1 in Proposition 3.2 with S = Xη the geometric generic fibre of
f : X → Y = Alb(X), and g equal to σ∗ : NS(X) → NS(X) restricted to the subspace
Λ := NS(X)|Xη of the space NS(Xη) = Pic(Xη).
Then, by Lemma 5.1, there is a nonzero v = V |Xη in Λ such that v is pseudo-effective,
v2 = 0 and g(v) = v. Moreover, again by Lemma 5.1, such v is unique up to a scalar and
v is nef. Now (σ∗V − V )|Xη = 0 in NS(Xη). By Lemma 2.14, replacing V by a multiple,
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 17
we may assume that V is a Cartier integral divisor on X and σ∗V = V + f ∗M where M
is a Cartier integral divisor on Y .
Then, as the divisor N(Xη) on Xη in Lemma 5.1, is uniquely determined by V , the
relation g(N(Xη)) ∼Q N(Xη) in Lemma 5.1 implies σ(N(Xη)) = N(Xη). It follows that
the Zariski closure of the image of N(Xη) in X is a σ-stable proper closed subset of
X . Thus N(Xη) = 0 since the wild automorphism σ has no non-empty closed proper
subset of X . Hence (KX + V )|Xη ∼ 0. Also (KX + V )|Xη ∼ 0. Now −KXη ∼ V |Xη
(like V |Xη = v) is nef with zero self-intersection. Hence κ(Xη,−KXη) ∈ {0, 1} by the
Riemann-Roch theorem.
If κ(Xη,−KXη) = 0, then σ stabilises the unique member in |−KXη | and its closure on
X , contradicting that σ is wild. If κ(Xη,−KXη) = 1, then a multiple of −KXη supports
fibres of the unique elliptic fibration on Xη, the set (and its closure in X) of singular
fibres of which is σ-stable. This contradicts that σ is wild. The claim is proved. �
This proves Proposition 5.2. �
6. Proof of Theorem 1.5
We begin with Proposition 6.1, which in turn requires the results between Propositions
6.2 and 6.5.
Proposition 6.1. Let X be a projective variety over C with a wild automorphism σ.
(1) Suppose dimX ≤ 3. Then X is either an abelian variety or a Calabi-Yau threefold
with the canonical divisor KX ∼ 0. Moreover σ has zero entropy.
(2) Suppose dimX = 4. Then σ has zero entropy, unless the Kodaira dimension
κ(X) = −∞ and the irregularity q(X) > 0.
We need the following results.
Proposition 6.2. Let X be a (necessaily smooth) projective variety over C. Suppose σ
is a wild automorphism of X. Put
A := ⊕k≥0H2k+1(X,Z)/(torsions), B := ⊕k≥0H
2k(X,Z)/(torsions).
Then the set of complex eigenvalues of σ∗|A and the set of complex eigenvalues of σ∗|B
coincide counted with multiplicities.
Proof. By Proposition 2.6, the topological Euler number e(X) = 0. Thus A and B have
the same rank, say r > 0. Let {ai}ri=1 be the set of complex eigenvalues of σ
∗ on A
counted with multiplicities and {bi}ri=1 be the set of complex eigenvalues of σ
∗ on B
counted with multiplicities.
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18 KEIJI OGUISO AND DE-QI ZHANG
Since σn is wild, the topological Lefschetz number of σn is 0, i.e.,∑
i
ani =∑
i
bni
for all n, in particular, for all n = 1, 2, ... , r.
Thus, the elementary symmetric function values of ai and bi coincide, hence the monic
polynomial with roots {ai}ri=1 counted with multiplicities and the one with roots {bi}
ri=1
counted with multiplicities coincide. Hence {ai}ri=1 = {bi}
ri=1 counted with multiplicities.
�
Corollary 6.3. Let X be a Calabi-Yau manifold of dimension three. Suppose σ is a wild
automorphism of X. Then KX is trivial in Pic(X) and σ has zero entropy.
Proof. Since χ(OX) = 0 by Proposition 2.6 and h1(OX) = 0 by assumption, it follows
that h3(OX) > 0. Hence h0(OX(KX)) = h
0(OX) > 0 by the Serre duality. Thus KX = 0
in Pic(X) as KX is Q-linearly equivalent to 0.
We show that σ has zero entropy. Indeed, by [4], replacing X by the universal cover and
σ by its lifting, we may assume that X is a Calabi-Yau threefold in the strict sense. Since
h1(OX) = 0, under the notation of Propoition 6.2, we have A = H3(X,Z)/(torsions). By
Proposition 6.2, it suffices to show that the complex eigenvalues of σ∗|A are roots of the
unity.
By the duality, we have b5(X) = b1(X) = 0. Hence the cohomology group H3(X,C)
is primitive. So is each Hodge component Hp,q(X) of H3(X,C). Thus the product on
H2,1(X) defined below
H2,1(X)×H2,1(X) ∋ (η, η) 7→ η ∧ η
is (up to a unit) a definite Hermitian form by the Hodge-Riemann relation as in [8, page
123]. Hence the action of σ∗ on H1,2(X) ⊕ H2,1(X) = H1,2(X) ⊕ H1,2(X) is unitary.
For the same reason, the action of σ∗ on H3,0(X) ⊕ H0,3(X) = H3,0(X) ⊕ H3,0(X) is
also unitary. So the eigenvalues of the action of σ on H3(X,C) are of absolute value 1.
Therefore, they are roots of the unity since the action of σ∗ is defined on the integral
H3(X,Z) and by Kronecker’s theorem. This proves the corollary. �
Corollary 6.4. Let X be a projective variety of dimension ≤ 4 such that the Kodaira
dimension κ(X) ≥ 0. Suppose σ is a wild automorphism of X. Then σ has zero entropy.
Proof. By Proposition 2.6, KX ∼Q 0 and we may assume that X coincides with its
Beauville-Bogomolov minimal split cover and that each factor Xi of X is stable under
σ so that σ|Xi is wild. Then χ(OXi) = 0 by Proposition 2.6 (1) and also dimXi ≤ 4.
Therefore Xi is either an abelian variety or a Calabi-Yau threefold in the strict sense
-
WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 19
(Proposition 2.6). Note that σ has zero entropy if and only if σ|Xi has zero entropy for
every factor Xi. Then the result follows from Corollary 6.3 and [20, Theorem 0.2]. �
Proposition 6.5. Let X be a projective variety of dimension four over C such that
q(X) = 0. Suppose σ is a wild automorphism of X. Then σ has zero entropy.
Proof. In what follows, we denote the free part Hk(X,Z)/(torsions) of Hk(X,Z) by
Hk(X,Z)f .
Let dk(σ) be the k-th dynamical degree of σ, that is, the spectral radius of σ∗|Hk,k(X,R).
It suffices to show the claim that d1(σ) = 1.
Let ai (1 ≤ i ≤ r) be the complex eigenvalues of σ∗|H2(X,Z)f counted with multiplicities.
Then the complex eigenvalues of (σ−1)∗|H2(X,Z)f are a−1i (1 ≤ i ≤ r) counted with multi-
plicities. Since H2(X,Z)f and H6(X,Z)f are dual under the cup product, the complex
eigenvalues of σ∗|H6(X,Z)f are also a−1i (1 ≤ i ≤ r) counted with multiplicities.
Then, by the Perron-Frobenius theorem, we have
1 ≤ d1(σ) = ai , 1 ≤ d3(σ) = d1(σ−1) = a−1j
for some 1 ≤ i, j ≤ r possibly i = j. If i = j, then d1(σ) = 1. This is because ai = 1
derived from 1 ≤ ai and 1 ≤ a−1i if i = j.
So, from now on we may assume that i 6= j. Then, by renumbering ai (1 ≤ i ≤ r) and
by interchanging σ and σ−1 if necessary, we may assume that
d1(σ) = a1 ≥ a−1r = d1(σ
−1) = d3(σ) ≥ 1.
Recall that b7(X) = b1(X) = 0 by the assumption. Then, by Proposition 6.2, a1 is also
an eigenvalue of either σ∗|H3(X,Z)f or σ∗|H5(X,Z)f .
Assume first that a1 is an eigenvalue of σ∗|H3(X,Z)f . Choose an eigenvector 0 6= u ∈
H3(X,C). Since σ∗ preserves the Hodge decomposition, we may choose u so that u ∈
Hp,q(X) in some Hodge componentHp,q(X) ofH3(X,C). Consider the complex conjugate
u of u with respect to H3(X,R)⊗R C = H3(X,C). Then u ∈ Hq,p(X) and σ∗u = a1u, as
σ∗ is defined over H3(X,R) and a1 is real.
On the other hand, H3(X,Z) is primitive by b7(X) = b1(X) = 0. Hence the Hodge
component Hp,q(X) is also primitive. Then, as in the proof of Corollary 6.3, there is a
complex number cp,q such that
cp,q(v.v.η) > 0
for every v ∈ Hp,q(X) \ {0} and for every Kähler class η ∈ H1,1(X,R) by the Hodge-
Riemann relation for the primitive cohomology group Hp,q(X) ([8, page 123]). In partic-
ular,
uu 6= 0
-
20 KEIJI OGUISO AND DE-QI ZHANG
in H3,3(X,R) and satisfies
σ∗(uu) = a21uu.
Thus, a21 is a positive real eigenvalue of σ∗|H3,3(X,R). Since d3(σ) = a
−1r is the spectral
radius of σ|H6(X,R), it follows that
a21 ≤ d3(g) = a−1r ≤ a1.
For the last inequality, we used our assumption that a1 ≥ a−1r ≥ 1. Hence a1 = 1 again,
since 1 ≤ a1. Thus d1(σ) = 1 as claimed.
Next we consider the case where a1 is an eigenvalue of σ∗|H5(X,Z)f . Then a1 is an
eigenvalue of (σ−1)∗|H3(X,Z)f by the duality. Then applying the same argument as above
for σ−1, we find that there are a Hodge component Hp,q(X) of H3(X,C) and a non-zero
element u ∈ Hp,q(X) such that uu ∈ H3,3(X,R) is an eigenvector of (σ−1)∗|H3,3(X,R)
corresponding to an eigenvalue a21, which is real and positive. Thus,
a21 ≤ d3(σ−1) = d1(σ) = a1.
Since 1 ≤ a1, it follows that a1 = 1. Hence d1(σ) = 1 also in this case, as claimed.
This proves the proposition. �
6.6. Proof of Proposition 6.1. By Theorem 3.1 and Propositions 3.2, 4.1 and 5.2, we
already know that a projective variety X with dimX ≤ 3 and with a wild automorphism
is isomorphic to either an abelian variety or a Calabi-Yau threefold. Thus Proposition
6.1 (1) follows from Corollary 6.3. Proposition 6.1 (2) follows from Corollary 6.4 and
Proposition 6.5.
We need the following to complete the proof of Theorem 1.5.
Proposition 6.7. Let X be a projective variety of dimX = n over C with a wild auto-
morphism σ. Then the albanese morphism albX : X → Alb(X) is a smooth surjective
morphism with connected fibres and hence 0 ≤ q(X) = dimAlb(X) ≤ n. If q(X) = n,
then X = Alb(X), an abelian variety; if q(X) ≥ n− 1, then σ has zero entropy.
Proof. By Lemma 2.7, albX : X → Alb(X) is a smooth surjective morphism with con-
nected fibres. Assume that q(X) = n−1. Then, since the fibre of albX is one dimensional,
d1(σ|X) = d1(σ|Alb(X)) = 1
by the product formula in [4] and the fact that σ|Alb(X) is a wild automorphism of an
abelian variety and hence has zero entropy (cf. Lemma 2.5). The rest of the proposition
is clear. �
6.8. Proof of Theorem 1.5. The theorem follows from Propositions 6.1 and 6.7.
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WILD AUTOMORPHISMS OF PROJECTIVE VARIETIES 21
7. Wild automorphisms of zero entropy
In this section, we consider the case where X is a (smooth) projective variety with a
wild automorphism σ such that σ is of zero entropy.
Lemma 7.1. Let σ be an automorphism of zero entropy of a smooth projective variety.
Then the following three conditions are equivalent.
(1) σ is wild.
(2) The twisted homogeneous coordinate ring B(X,L, σ) is a projectively simple for
any σ-ample line bundle L on X.
(3) The twisted homogeneous coordinate ring B(X,L, σ) is a projectively simple for
at least one ample line bundle L on X.
Proof. By [9, Theorem 1.2 (2)], any ample line bundle on X is σ-ample, because σ is of
zero entropy. Thus, the implication (2) ⇒ (3) is clear and the implications (1) ⇒ (2) and
(3) ⇒ (1) follow from [20, Proposition 0.1]. �
Lemma 7.2. Let X be a (smooth) projective variety. Suppose σ is a wild automorphism
of X of zero entropy. Then σ∗|NSC(X) is not diagonalizable, unless X is isomorphic to an
abelian variety.
Proof. Since σ∗|NSC(X) is defined over NS(X), the eigenvalues of σ∗|NSC(X) are all roots of
unity. So, if σ∗|NSC(X) is diagonalizable, then (σ∗)m = id on NS(X) for some m > 0. In
this case, X is an abelian variety by Corollary 2.4. This proves the lemma. �
The following conjecture is in [18, Question 2.6].
Conjecture 7.3. (cf. [18]) Every nef Q-Cartier divisor (not necessarily effective) on a
Calabi-Yau manifold of dimension three is Q-linearly equivalent to an effective divisor.
The following is asserted in [10].
Theorem 7.4. (cf. [10, Theorem 4.7]) Assume Conjecture 7.3 for the Calabi-Yau man-
ifold X of dimension three. Then X does not admit a wild automorphism.
Remark 7.5. In the proof of [10], he assumes that the (smooth) Calabi-Yau threefold
X is simply connected. One can assume so, because one can lift a wild automorphism σ
on a Calabi-Yau manifold to its (finite) universal cover. Apparently, his proof implicitly
assumes that σ is of zero entropy (one can do so since Proposition 6.1 above is available
now) and implicitly uses the result in Lemma 7.6 below, which may not hold when σ is
of positive entropy. For the sake of completeness, we supply proofs of Lemma 7.6 and
Theorem 7.4 below.
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22 KEIJI OGUISO AND DE-QI ZHANG
Lemma 7.6. Let X be a normal projective variety and σ a automorphism. Suppose
σ∗|NSC(X) is of infinite order and σ is of zero entropy. Then there is a nef (integral)
Cartier divisor L 6≡ 0 such that σ∗L ≡ L (numerical equivalence) possibly after replacing
σ by a positive power. In particular, we may assume σ∗L ∼ L if the irregularity q(X) = 0.
Proof. Replacing σ by a positive power, we may assume that σ∗|NSC(X) is unipotent (but
not trivial) so it is equal to I +N with N nilpotent. Let m (≥ 2) be the maximal order
of Jordan blocks of σ∗|NSC(X). Then Nm = 0, but Nm−1 6= 0. Thus
g := lims→∞
(σ∗)s/sm−1 = Nm−1/(m− 1)!
is a nonzero linear transformation on NSQ(X). Since ample Cartier divisor classes gener-
ate NSQ(X) and since g = Nm−1/(m− 1)! 6= 0 on NSQ(X), there is an ample Cartier di-
visor H on X so that L := g(H) is nonzero Q-Cartier divisor. Since Ls := (σ∗)s(H)/sm−1
is ample, L = lims→∞ Ls is nef. Moreover, using σ∗ = I +N and Nm = 0, we compute
σ∗L = lims→∞
(I +N)s+1(H)/sm−1 = (Nm−1/(m− 1)!)(H) = g(H) = L.
The lemma follows by replacing L by a positive multiple. �
7.7. Proof of Theorem 7.4. In view of Proposition 6.1, we consider a wild automor-
phism σ of zero entropy on a smooth Calabi-Yau threefold X (in the strict sense, and
want to reach a contradiction).
By Lemma 7.6, we have an integral nef Cartier divisor L 6≡ 0 such that σ∗L ∼ L. Since
we assume that Conjecture 7.3 holds on X , it follows that |mL| 6= ∅ for some positive
integer m, contradicting Proposition 2.6 (2). This proves Theorem 7.4.
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24 KEIJI OGUISO AND DE-QI ZHANG
Department of Mathematical Sciences, the University of Tokyo, Meguro Komaba 3-8-
1, Tokyo, Japan, and National Center for Theoretical Sciences, Mathematics Division,
National Taiwan University Taipei, Taiwan
E-mail address : [email protected]
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge
Road, Singapore 119076, Republic of Singapore
E-mail address : [email protected]
1. Introduction2. Preliminary results in arbitrary dimensions3. Preliminary results in lower dimensions4. Proof of Theorem 1.2 when q = 25. Proof of Theorem 1.2 when q = 16. Proof of Theorem 1.57. Wild automorphisms of zero entropyReferences